I'm stuck with the following part of an exercise that asks to proof Eisenstein's Criterion:
Let $P$ be a polynomial over $\mathbb{Q}$ of the form $ P = a_nX^n + a_{n-1}X^{n-1} + .... + \ a_0 $ such that $ p \nmid a_n$, $p \mid a_j \ (\forall i \in {0,...,n-1})$ and $p^2 \nmid a_0$. Now suppose that there are two polynomials $Q$ and $R$ such that $P = Q * R$ and neither $Q$ nor $R$ are euqal to $\pm 1$. Show that both $Q$ and $R$ are of degree bigger or euqal to $1$.
I attempted the following:
W.l.o.g. we suppose that $Q$ is a constant polynomial meaning $R = c_0$. We further set $Q$ to $ Q = b_nX^n + b_{n-1}X^{n-1} + .... + \ b_0 $. Therefore we got:
$ P = c_0b_nX^n + c_0b_{n-1}X^{n-1} + .... + \ c_0b_0 $
Since $ p \nmid a_n$ we know that $ p \nmid c_0$ and $p \nmid b_n$. And since for all $i$ holds $a_i = c_0*b_i$ it follows from $p \mid a_j \ (\forall j \in {0,...,n-1})$ that $ p \mid b_j \ (\forall j \in {0,...,n-1})$.
But I don't know how to continue from here. Could you give me any hints?
Thanks in advance.
The statement is obviously false. Just take $Q(x)=P(x)$ and $R(x)=1$.